![](https://cv01.studmed.ru/view/7b1fefb4e81/bg72.png)
After substituting Eq. (3.39) into Eq. (3.38), we obtain the governing
equation of motion
(3.40)
where K
o
M is the added mass due to the fluid. From Eq. (3.40), we see that
placing a single degree-of-freedom system in a fluid increases the total mass
of the system and adds viscous damping to the system.
9
A practical applica-
tion of the fluid mass loading is in modeling offshore structures.
10
In this section, the use of force-balance and moment-balance methods for
deriving the governing equation of a single degree-of-freedom system was il-
lustrated. In the next section, a different method to obtain the governing equa-
tion of a single degree-of-freedom system is presented. This method is based
on Lagrange’s equations, where one makes use of scalar quantities such as ki-
netic energy and potential energy for obtaining the equation of motion.
3.6 LAGRANGE’S EQUATIONS
11
Lagrange’s equations can be derived from differential principles such as the
principle of virtual work and integral principles such as those discussed in
Chapter 9. We will not derive Lagrange’s equations here, but use these equa-
tions to obtain governing equations of holonomic
12
systems. We first present
Lagrange’s equations for a system with multiple degrees of freedom and
then apply them to vibratory systems modeled with a single degree of free-
dom. In Chapter 7, we illustrate how the governing equations of systems with
multiple degrees of freedom are determined by using Lagrange’s equations.
Let us consider a system with N degrees of freedom that is described by
a set of N generalized coordinates q
i
, i 1, 2, . . ., N. These coordinates are
unconstrained, independent coordinates; that is, they are not related to each
other by geometrical or kinematical conditions. Then, in terms of the chosen
generalized coordinates, Lagrange’s equations have the form
(3.41)
where are the generalized velocities, T is the kinetic energy of the system,
V is the potential energy of the system, D is the Rayleigh dissipation function,
q
#
j
d
dt
a
0T
0q
#
j
b
0T
0q
j
0D
0q
#
j
0V
0q
j
Q
j
j 1,2, . . . , N
1m K
o
M2
d
2
x
dt
2
C
f
dx
dt
kx f 1t 2
3.6 Lagrange’s Equations 93
9
A similar result is obtained when one considers the acoustic radiation loading on the mass, when
the surface of the mass is used as an acoustically radiating surface. See L. E. Kinsler and A. R.
Frey, Fundamentals of Acoustics, 2nd ed., John Wiley and Sons, NY, pp.180–183 (1962).
10
A. Us´cilowska and J. A. Kol
´
odzeij, “Free vibration of immersed column carrying a tip mass,”
J. Sound Vibration, Vol. 216, No. 1, pp. 147–157 (1998).
11
For a derivation of the Lagrange equations see D. T. Greenwood, Principles of Dynamics, Pren-
tice Hall, Upper Saddle River, NJ, 1988, Section 6-6.
12
As discussed in Chapter 1, holonomic systems are systems subjected to holonomic constraints,
which are integrable constraints. Geometric constraints discussed in Chapter 1 are in this category.